document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189659>Question 1189659</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A particle moves in a straight line with velocity v(t)=root(3t-1) metres per second where t is time in seconds.At t=2, the particle's distance from the starding point was 8 meters in the positive direction. What is the particle's position at t=7seconds? </SPAN>\n" );
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document.write( "To find the position function s(t), we integrate the velocity function v(t)\r<BR>\n" );
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document.write( "Recall that the velocity is the derivative of the position function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large v(t) = \frac{d}{dt}s(t)\">\r<BR>\n" );
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document.write( "so the integral will reverse this process\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(t) = \int v(t)dt\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(t) = \int \sqrt{3t-1}dt\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(u) = \frac{1}{3}\int \sqrt{u}du\"> Let u = 3t-1, so du = 3dt which means dt = (1/3)du\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(u) = \frac{1}{3}\int u^{1/2}du\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(u) = \frac{1}{3}*\frac{1}{1+1/2}u^{1+1/2}+C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(u) = \frac{1}{3}*\frac{1}{3/2}u^{3/2}+C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(u) = \frac{1}{3}*\frac{2}{3}u^{3/2}+C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(u) = \frac{2}{9}u^{3/2}+C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(t) = \frac{2}{9}(3t-1)^{3/2}+C\"> Plug in u = 3t-1\r<BR>\n" );
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document.write( "To verify we have the correct antiderivative, differentiate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{2}{9}(3t-1)^{3/2}+C\"> with respect to t, and you should get <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large v(t) = \sqrt{3t-1}\"> again.\r<BR>\n" );
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document.write( "Plug in the condition that s(2) = 8 and solve for C.<BR>\n" );
document.write( "In other words, plug in t = 2 and s(t) = 8.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(t) = \frac{2}{9}(3t-1)^{3/2}+C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(2) = \frac{2}{9}(3*2-1)^{3/2}+C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 8 \approx 2.48451997 +C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 8-2.48451997 \approx C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 5.51548003 \approx C\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large C \approx 5.51548003\">\r<BR>\n" );
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document.write( "This particular position function is approximately<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(t) \approx \frac{2}{9}(3t-1)^{3/2}+5.51548003\">\r<BR>\n" );
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document.write( "Now plug in t = 7 to find the position of the particle at the time of 7 seconds\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(t) \approx \frac{2}{9}(3t-1)^{3/2}+5.51548003\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(7) \approx \frac{2}{9}(3*7-1)^{3/2}+5.51548003\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s(7) \approx 25.39163983\">\r<BR>\n" );
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document.write( "The particle is <font color=red>roughly 25.39163983</font> meters from the starting point at t = 7 seconds. Round this value however you need to.<BR>\n" );
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